From the letters of the word “PAINTERS”, all possible 8-letter arrangements are formed. Find the denominator of the probability, in simplest form, that all vowels always come together.

ATQ, The word “PAINTERS” has 8 different letters. Vowels are A, I and E. Number of vowels = 3 Number of consonants = 5 Total arrangements of 8 different letters = 8! For favourable arrangements, all vowels must come together. Treat the 3 vowels as one block. Now total units = 1 vowel block + 5 consonants = 6 units These 6 units can be arranged in 6! ways. The 3 vowels inside the block can be arranged in 3! ways. Favourable arrangements = 6! × 3! Probability = Favourable arrangements/Total arrangements Probability = 6! × 3!/8! Probability = (6! × 6)/(8 × 7 × 6!) Probability = 6/56 Probability = 3/28 So, the denominator of the probability in simplest form is 28.